# Comparing Return Probabilities

# Comparing Return Probabilities

Consider a Cayley graph of a group, $\Gamma$. Suppose that $W$ is a random assignment of nonnegative numbers to the edges and that the law of $W$ is $\Gamma$-invariant. Let $X_t$ be continuous-time random walk on $\Gamma$ in the random environment $W$: incident edges $e$ are crossed at rate $W(e)$. Write $p^W(t) := {\bf E}\bigl[{\bf P}_o[X_t = o]\bigr]$ for the expected return probability at time $t$ (averaged over $W$). Fontes and Mathieu asked whether given two such environments, $W_1$ and $W_2$, with $W_1(e) \le W_2(e)$ for all edges $e$, one has $p^{W_1}(t) \ge p^{W_2}(t)$ for all $t \ge 0$. When the pair $(W_1, W_2)$ has a $\Gamma$-invariant law, this was shown by Aldous and the speaker. It remains open in general. We attempt toattack this problem via similar questions for finite graphs.